I'm having a lot of trouble with this topic. I have no idea on what to do for these questions:

1) Show that the set (0)=1\}" alt="S=\{p\in\mathbb{P}_2(0)=1\}" /> is NOT a subspace of

The only thing I can think of doing for this question is making

You want to show that the given set is not a subspace of . Recall the definition of a vector subspace. You need to show that one of the vector subspace axioms does not hold in this case; that is, you need to give a counter example to one of the axioms.

You want to show that the given set is not a subspace of . Recall the definition of a vector subspace. You need to show that one of the vector subspace axioms does not hold in this case; that is, you need to give a counter example to one of the axioms.

No. You need to show that the given set is a subspace of . Again, recall the axioms of vector space and show that they hold for any polynomials in your set S.

I'll start with closure to addition: Let . Then by definition of S. Now, note that , that is, and so by definition of S,

Can you do the rest?

For the first question, do you because do we say that the zero vector is not included and therefore the set is empty.

For the first question, do you because do we say that the zero vector is not included and therefore the set is empty.

Since for any then the zero polynomial is not in S. This, of course, does not mean that S is empty. But one of the subspace axioms is that the space must have the identity element. Here it does not, therefore S is not a subspace.

Since for any then the zero polynomial is not in S. This, of course, does not mean that S is empty. But one of the subspace axioms is that the space must have the identity element. Here it does not, therefore S is not a subspace.

Can you think of any polynomial such that ? (hint: its simple).

Okay, so would I just prove that a degree one polynomial is a subspace of ?